Charge separation on colliding superconducting cosmic strings

Volume 250, number 1,2

PHYSICSLETTERSB

1 November 1990

Charge separation on colliding superconducting cosmic strings T.J. Allen Department of TheoreticalPhysics, Research School of Physical Sciences, Australian National University, Canberra, ACT, Australia

Received 2 July 1990

A mechanismwhereby superconducting cosmicstrings can acquire large net charges is explored. This has a significant bearing on superconductingcosmicstring cosmologiesbecause charged stringscan form long-livedstableloops which are largerthan loops stabilized purely by current.

1. Introduction

2. Collisions

It has been shown that a superconducting cosmic string bearing a current [ 1 ] may be stable against collapse and thus long-lived. The current needed to stabilize a string loop at a particular radius is proportional to the radius of the loop. Because current quenching provides an upper bound to the current which may be carried by a superconducting cosmic string, there is thus an upper bound to the radius of a loop which may be stabilized by the current. For some theories this radius might be quite small and macroscopic loops would not form. However, it has also been shown [2,3 ] that a charge can stabilize a string. Unlike the current, the charge is not limited by quenching. In fact, Davis and Shellard have shown [2 ] that a charged string actually has a greater capacity to carry a current. A charge-stabilized string, then, may be much larger than one stabilized only by the current, and must significantly affect cosmology. It was demonstrated by Davis and Shellard [4] that, even in the absence of primordial magnetic fields, random fluctuations must produce currents and charge distributions capable of stabilizing microscopic strings, or vortons. The mechanism explored in the present work, by contrast, provides currents established by primordial magnetic fields the potential to create and maintain charges large enough to stabilize macroscopic strings.

Charge separation may occur when superconducting cosmic strings collide because the centre-of-momentum reference frame of the collision will not in general be the same as the rest frames of the colliding strings. A string which may have no charge in its rest frame will in general possess a charge distribution in some other frame. At the collision the strings intercommute [ 5 ] enabling the local charge distributions to produce net global charges on the new strings formed after the collision. The principle will be demonstrated here by means of an example which embodies all the relevant features. A circular loop, stabilized by a current alone, collides with a long straight piece of string, which may be part of a larger loop. The strings intersect at two points, as shown in fig. 1. In order to avoid having to model the dynamics of kinked superconducting cosmic strings it is necessary to work in a frame where the two collisions are simultaneous. If the collisions are not simultaneous, leakage of charge will occur, which manifests itself in the lack of Lorentz invariance of the separated charge. At the time of collision, each string fragments at the intersection points then reconnects with the other string. The charge on each fragment will give the total charge on the new strings. Although some charge will likely be lost in the collision, this charge loss will presumably be local to the intersection points and hence not greatly affect the total macroscopic charge for

0370-2693/90/$ 03.50 © 1990 - ElsevierScience Publishers B.V. ( North-Holland )

29

Volume 250, number 1,2

PHYSICS LETTERS B

1 November 1990

p~(y~) =0, 2nR 0

0"2

2 . a2 2 R)d(Yo2-Rsln~)d(Y03, .

×6(y~,-Rcos

(2) Here y~; refers to t h e / t h component of a general position in the rest-frame of the jth string. The collision is arranged by boosting each string separately to a c o m m o n frame, the first by a velocity

v~= (0, O, v3)

(a)

and the second by •2= (Vl, O, O) ,

with Lorentz factors

71=[1--(V3)2] -1/2 and 72 = [1 - (v l )21 --1/2 In the frame of the collision the string positions are now x, ( a , ) = ( a a , , O, - v 3 t )

(b)

Intersection

Fig. 1. Configuration of the collision. (a)Before collision. (b) Immediately after instant of collision.

,

occurs when

t=0,

a2=O, nR

and

a, = +_R/oty2. The boosted charges and currents become

pl(y,)=O, sufficiently long strings. The rest-frame positions of the strings are

Jl (y,) = f da, ( a j , , 0, 0)

Z 1( a, ) = ( a a , , 0, 0 ) ,

Z2(e2)=R(cosa2,.sin -~-, a2 0 ) \

R

X 6(y it - ota, )6(y12)6(7i-'Y~3 '

0 < e 2 ~2rtR

"

p~(J,~) = 0 ,

d~(y~) = f de, (otj,, 0, 0)d(y~, ×6(y~2)6(y~s), 30

2rtR

(1)

They carry rest-frame current and charge densities

v3t),

P~(Y,)=72v,j2

J da2 sin a2 R o

X(j(y21-Rcos~)d(y22-Rsin~)(j(y23, -ore, )

,

(4)

1 November 1990

PHYSICS LETTERSB

Volume 250, number 1,2 2~rR

lisions would be preserved because the collisions are separated by a displacement purely in the x-direction. The transformed charges would be

, 2 ( y l ) = f da2J2 ( - ~,2sin ~ , cos --~, 0) o

×6(y2,-RCOSR)6(y22-Rsin-~)6(y23)

.

p;(y2)=o,

(4 cont'd)

pZ(y2) =~,3p2 ( y , ) ,

while there is no charge on the straight string, the circular (elliptical now, actually) string has acquired a local charge distribution which becomes a global charge when the string fragments and intercommutes with the other string. This is illustrated in fig. 2. The charge and current densities are now specified as functions of general position coordinates in the new common frame, Yl. The jacobian

giving a separated charge

Oyl -1 IJl=~y~=Y2

p3=(V31 ' 0, O)

(6)

Oy2

e'sep = f d3y2p2(y2)=)/3 f d3yl Oy I pl2(yl)

= Qsep, since the jacobian I 0y2/0yl I = (Y3)-1. However, if the boost was

with enables integration of the total charge with respect to the new reference frame. The total charge separated is nR

QseP=fd3ylp~=vlJ2f d~2sina--!2R 0

Y3= [1 - (v3t)2] - 1 / 2 the collisions would no longer be simultaneous, but would be separated by a time difference At = 2Ry3 (y2) -i. The transformed charges would be p l ( y 2 ) = --O£y3v31Jl f d°'l ~ ( y l l -oLo" 1)

=2VlJ2R .

(5)

This separated charge is invariant under Lorentz boosts which preserve the simultaneity of the collisions, but not under boosts which do not. If a further boost by a velocity

X ~(yI2)~(y~3) , 2nR

p2(y2)=~3y2J2(Vl +V31)

3 0

. a2 dcr2 s m

/-y3m-~(0, V32 , V33 )

with

(7)

~Y3= [1 - (v32) 2 - (v33) 2 ] -1/2

giving a separated charge of

were applied to both strings, simultaneity of the col-

Q'sep=2Rj2(vl +v31 ) - -2j1Rv31 71 72

B

+ Fig. 2. Charge is separated whenthe strings intercommute.

(8)

This tells us that unless the collisions are simultaneous it is necessary to dynamically model the leakage of charge that occurs between the two collisions. In order to see how large the invariant charge separation is, it is useful to compare it with the charge required to stabilize a circular string at the radius of the colliding circular string, without current. It was demonstrated in ref. [ 3] that for the circular string stabilized by pure current in its rest-frame the magnitude of the current must be such that J2 = q x / ~ , 31

Volume 250, number 1,2

PHYSICS LETTERS B

where q is the charge of each of the charge carriers on the string and T is the string tension. It is found then that the separated charge is Qsep= v~qRx~. To stabilize a loop of radius R with charge requires a charge density 2nR

P=

f d°'qx//~T~3(y-Z(tT)) o

giving a total charge Qstab :

~

d3yp=nqRx/~"

(9)

The separated charge is thus Qsep = Vl Qstab • 7[

So, although the charge separated is not, in this case, large enough to stabilize the original string, it is clearly still quite large when the strings collide at a large relative velocity and would be large enough to stabilize a loop with radius VlR/n.

1 November 1990

may be expected to retain their charge. The inertia of the charge carrying modes on the string would tend to round out structure. Arguing by analogy with the circular case, for a string with charge jo and current IJ[ locally, structure with a radius of curvature less than R = [ (J°) 2+,/2 ] / q x ~ would be rounded out [ 3 ]. Cusps on ordinary cosmic strings can occur because the string tension equals the mass per unit length. For superconducting cosmic strings, however, the effective string tension is decreased by the presence of current or charge [ 1 ], so cusp formation would be suppressed. Emission of electromagnetic radiation would also tend to smooth out structure on strings, so the tendency would be to form stable loops.

Acknowledgement It is a pleasure to thank Lindsay Tassie for many helpful discussions and much useful advice in the preparation of this manuscript.

References 3. Conclusions The evolution of a network of superconducting cosmic strings would clearly involve large charges being continually exchanged. The consequences of this for the early universe would require modification of the original Ostriker-Thompson-Witten [ 6 ] explosive structure formation scenario. Large charged stable loops may be formed after loop fragmentation has proceeded to the point where collisions are no longer common and hence most strings

32

[ 1 ] E. Copeland, M. Hindmarsh, D. Haws and N. Turok, Nucl. Phys. B 306 (1988) 908. [2] R.L. Davis and E.P.S. Shellard, Phys. Lett. B 209 (1988) 485. [3] T.J. Allen, Phys. Lett. B 231 (1989) 429. [4] R.L. Davis and E.P.S. Shellard, Nucl. Phys. B 323 (1989) 209. [5] P. Laguna, Proc. Yale Cosmic String Workshop, eds. F.S. Accetta and L.M. Krauss (World Scientific, Singapore, 1988) p. 42. [6] J. Ostriker, C. Thompson and E. Witten, Phys. Lett. B 180 (1986) 231.